Q: A centrifuge spins at $$7.5 \times 10^4$ revolutions per minute. What is its angular velocity in radians per second?

$$\omega = 7.85 \times 10^3$ rad/s

Question 1

A figure skater spins with her arms extended and then pulls them in close to her body. Which of the following statements best describes what happens to her angular velocity and moment of inertia, assuming no external torque acts on her?

Accepted Answer

Her angular velocity increases and her moment of inertia decreases.

Question 2

A solid cylinder of mass $$m$$ and radius $$R$$ rolls without slipping down an incline of height $$h. $$What is its speed at the bottom of the incline?

Accepted Answer

$$v = \sqrt{\frac{2gh}{1 + \frac{1}{2}}}$$

Question 3

A wind turbine blade is modeled as a thin rod of length $$L$$ and mass $$M$$ rotating about one end. What is its moment of inertia?

Accepted Answer

$$I = \frac{1}{3}ML^2$

Question 4

A car of mass $$m$$ travels around a flat curve of radius $$r at $$speed $$v. $$What is the required centripetal force to keep the car moving in a circle?

Accepted Answer

$$F_c = \frac{mv^2$}{r}$$

Question 5

A Ferris wheel rotates at a constant speed. At the bottom of the wheel, what is the direction of the net force acting on a passenger?

Accepted Answer

Upward, toward the center of the wheel

Question 6

A ball of mass $$m is $$attached to a string and swung in a horizontal circle of radius $$r at $$constant speed $$v. $$What is the magnitude of the centripetal acceleration?

Accepted Answer

$$a_c = \frac{v^2$}{r}$$

Question 7

A torque is applied to a door at a distance $$r$$ from the hinges with a force $$F at an $$angle $$	heta to $$the door. What is the magnitude of the torque?

Accepted Answer

$$	au = rF\sin	heta$$

Question 8

A merry-go-round of moment of inertia $$I is $$initially at rest. A child of mass $$m$$ runs tangentially at speed $$v$$ and jumps onto the edge at radius $$R. $$What is the final angular velocity $$\omega_f of $$the system?

Accepted Answer

$$\omega_f = \frac{mRv}{I + mR^2$}$$

Question 9

A solid cylinder and a hollow pipe have the same mass and radius and rotate about their central axes with the same angular velocity. Which has the greater rotational kinetic energy?

Accepted Answer

The hollow pipe

Question 10

A centrifuge spins at $$7.5 	imes 10^4$ revolutions per minute. What is its angular velocity in radians per second?

Accepted Answer

$$\omega = 7.85 	imes 10^3$ rad/s

Question 11

A disc of mass $$m$$ and radius $$R is $$thrown so that it both translates and rotates. If $$\frac{9}{10} of $$its kinetic energy is translational and $$\frac{1}{10} is $$rotational, what is the total kinetic energy?

Accepted Answer

$$K_{total} = \frac{1}{2}mv^2$ + \frac{1}{2}I\$$omega^2$$

Question 12

A force $$F is $$applied at a distance $$r$$ from the axis of rotation, but not perpendicular to the lever arm. Which component of the force produces the torque?

Accepted Answer

The component perpendicular to the lever arm

Question 13

A child pushes a merry-go-round with a force $$F at $$the edge (radius $$R). If $$the moment of inertia is $$I$$, what is the angular acceleration $$\alpha$$ produced?

Accepted Answer

$$\alpha = \frac{FR}{I}$$

Question 14

A solid cylinder of mass $$m$$ and radius $$R$$ rolls without slipping. What is its rotational kinetic energy when rotating at angular velocity $$\omega$$?

Accepted Answer

$$K_{rot} = \frac{1}{2}I\$$omega^2$$ = \frac{1}{2} \left(\frac{1}{2}mR^2$\right)\$$omega^2$$

Question 15

A spinning ice skater pulls her arms in, reducing her moment of inertia from $$I_1 to$$ $$I_2. If $$her initial angular velocity is $$\$$omega_1$$, what is her final angular velocity $$\$$omega_2$$?

Accepted Answer

$$\$$omega_2$$ = \frac{I_1$}{I_2$}\$$omega_1$$

Question 16

A thin rod of length $$L$$ and mass $$M$$ rotates about its center. What is its moment of inertia?

Accepted Answer

$$I = \frac{1}{12}ML^2$

Question 17

A particle moves in a circle of radius $$r at $$constant speed $$v. $$What is its angular velocity $$\omega$$?

Accepted Answer

$$\omega = \frac{v}{r}$$

Question 18

A rotating object has angular momentum $$L = I\omega. If no $$external torque acts, what happens to $$L$$?

Accepted Answer

It remains constant

Question 19

A solid cylinder and a box of equal mass start from rest at the top of two identical inclines. The cylinder rolls without slipping, while the box slides down a frictionless incline. Which reaches the bottom first?

Accepted Answer

The box

Chapter 9: Rotational Motion – Study Notes

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